On the second smallest prime non-residue

Abstract

Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. Let $q_1<q_2$ denote the two smallest prime non-residues of $\chi$. We give explicit upper bounds on $q_2$ that improve upon all known results. We also provide a good upper estimate on the product $q_1 q_2$ which has an upcoming application to the study of norm-Euclidean Galois fields.